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Linear Algebra I Proof (HW Help)

  1. Oct 22, 2012 #1
    1. The problem statement, all variables and given/known data

    So the question is, Prove the following:

    Let A be an n x n matrix. If there exists a vector v in Rn that is not a linear
    combination of the columns of A, then at least one column of A is not a pivot column.

    2. Relevant equations

    The only relevant theorem I think is the invertible matrix theorem, which i attached.
    I also attached theorem 4 (book has different names)


    3. The attempt at a solution

    So far, I started with

    - Let A be a n x n matrix and v be a vector in Rn that is not a linear combination of the columns of A
    - then there is not a pivot position in every row of rref A (theorem 8 not g to not c)
    - then there is at most n-1 pivot positions (out of n rows)
    - then at least one column of rref A is not a pivot position (square matrix).

    the question i have is, am i allowed to jump from my first sentence to my second sentence without any justification?

    For some reason, I am thinking that I'm supposed to go from (theorem 4 not b to not a)
    and then, since theorem 4a is equivalent to theorem 8g, then jump from (theorem 8 not g to not c). But am i allowed to use and jump from theorem 4 to theorem 8? since theorem 4 is for a m x n matrix, while theorem 8 is n x n matrix?

    Also, in theorem 4 there's a statement " each b in Rm is a linear combination of the columns of A", which is the assumption i started with. What would be the equivalent statement in theorem 8, if there is any?

    Sorry, this is probably a basic proof question, but I'm just horrible at proving something, so I wanted to make sure I was on the right path.
     

    Attached Files:

    Last edited: Oct 22, 2012
  2. jcsd
  3. Oct 22, 2012 #2
    What about span?
     
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